(i) Find the torque of a force `7hati + 3hatj - 5hatk` about the origin. The force acts on a particle whose position vector is `hati - hatj + hatk`.
(ii) Show that moment of a couple does not depend on the point about which you take the moments.

Explanation

(i) Here, `vecF = (7hati + 3hatj - 5hatk), vecr = (hati - 7hatj + hatk)`
`vectau = vecF =(hati - hatj + hatk) xx (7hati + 3hatj - 5hatk)`

`=|{:(hati,hatj,hatk),(1,-1,1),(7,3,-5):}| =hati(5-3)-hatj(-5-7)+ hatk(3+7)`
i.e., `vectau=(2hati + 12hatj + 10hatk)`
(ii) Moment of the copuler `(vecF,-vecF)` about an arbitrary point O, i.e., `vectau` = sum of the moment of `vecF` and `-vecF` about O.
`=vecr_(1) xx (-vecF) + vecr_(2) xx vecF = vecr_(2) xx vecF - vecr_(1) xx vecF`
From `Delta` law of vectors, `vecr_(1) + vec(AB) = vecr_(2)`
or `vec(AB) =(vecr_(2)-vecr_(1))`.......(ii)
From eqns. (i) and (ii),
`vectau = vec(AB) xx vecF`
From eqn, (iii), it follows that `vectau` is independent of the position of the point about which its moment is considered.